Systematic-Error-Tolerant Multiqubit Holonomic Entangling Gates

Quantum holonomic gates hold built-in resilience to local noises and provide a promising approach for implementing fault-tolerant quantum computation. We propose to realize high-fidelity holonomic $(N+1)$-qubit controlled gates using Rydberg atoms confined in optical arrays or superconducting circuits. We identify the scheme, deduce the effective multibody Hamiltonian, and determine the working condition of the multiqubit gate. Uniquely, the multiqubit gate is immune to systematic errors, i.e., laser parameter fluctuations and motional dephasing, as the $N$ control atoms largely remain in the very stable qubit space during the operation. We show that $C_N$-not gates can reach the same level of fidelity at a given gate time for $N\le 5$ under a suitable choice of parameters, and the gate tolerance against errors in systematic parameters can be further enhanced through optimal pulse engineering. In the case of Rydberg atoms, the proposed protocol is intrinsically different from typical schemes based on Rydberg blockade or antiblockade. Our study paves an alternative way to build robust multiqubit gates with Rydberg atoms trapped in optical arrays or with superconducting circuits. It contributes to current efforts to develop scalable quantum computation with trapped atoms and fabricable superconducting devices.

Figure 1

(a) Level diagram. Two low-lying states $|0⟩$ and $|1⟩$ couple resonantly to an auxiliary state $|A⟩$ with Rabi frequencies ${\mathrm{\Omega }}_0$ and ${\mathrm{\Omega }}_1$. (b) State $|A⟩$ resonantly couples to dressed state $|B⟩$ (Rabi frequency $\sqrt{{\mathrm{\Omega }}_0^2+{\mathrm{\Omega }}_1^2}$) but not to dressed state $|D⟩$. (c) Two-body Ising (density-density) interaction between the control qubits (strength $J_c$), and between the control and target qubits (strength $J_t$).

Figure 2

Collective states of the control qubits (left panel), $M$ denoting the number of qubits initially in state $|B⟩$. Due to the strong Ising interaction, the collective states with multiple qubits in state $|A⟩$ are prohibited from excitation. When $J_t$ is large, a geometric gate can be realized through one-step operation. The gate trajectory on the Bloch sphere is shown on the right panel.

Figure 3

(a) Three-level interaction with $M\ne 0$ in Eq. (4). (b) Three-level interaction represented on the dressed-state basis.

Figure 4

Rydberg atom setting and many-body dynamics. (a) Spatial configuration of the Rydberg atom array. The target-control qubit distance is $d_i=3.8\mu \mathrm{m}$. The control atoms are distributed uniformly on a ring with the nearest-neighbor separation to be $d_c=2\mu \mathrm{m}$, and are driven homogeneously by the laser fields. (b) Evolution of the $C_N$-not gate fidelity based on the full (solid lines) and effective (circles) Hamiltonian simulations, showing excellent agreement. (c) Population dynamics of the collective states with different numbers of $|r⟩$, when implementing the $C_3$-not gate with four atoms. The strong RRI prohibits the multiple Rydberg states from excitation. The red dotted line denotes the population of the target atom in state $|r{⟩}_t$. Here ${\overline{\mathrm{\Omega }}}_c=2\pi \times 40\mathrm{MHz}$ and ${\mathrm{\Omega }}_t=2\pi \times 1\mathrm{MHz}$. Other parameters are given in the text.

Figure 5

(a) not gate fidelity versus relative errors in $T$ and ${\mathrm{\Omega }}_t$. The fidelity is improved significantly with OHQC. (b) not gate fidelity when varying Rydberg lifetime $\tau$ as well as $\delta {\mathrm{\Omega }}_t$. OHQC is robust and achieves higher fidelities when $|\delta {\mathrm{\Omega }}_t|/2\pi >30\mathrm{kHz}$. The NHQC has high fidelities when $\delta {\mathrm{\Omega }}_t$ is small.

Figure 6

Error tolerance of the multiqubit holonomic gate. Fidelities of the $C_N$-not gate with $N=2$ and $N=3$ when introducing errors in (c) ${\mathrm{\Omega }}_0(t)$, (d) $\mathrm{\Delta }(t)$, and (e) $\omega$, respectively. (f) $C_N$-not gate fidelity for different $N$. The solid and dotted lines show the fidelity without errors and when the parameter suffers 5% relative errors, correspondingly.

Figure 7

Gate robustness against the RRI fluctuation and finite temperature. (a) not gate fidelity versus relative errors of the RRI. The gate becomes robust with increasing ${\overline{\mathrm{\Omega }}}_c/2\pi$ (labeled in the figure, unit MHz). (b) not gate fidelity decreases gradually when increasing atomic temperature $T_a$. Larger ${\mathrm{\Omega }}_{\max }/2\pi$ (labeled in the figure, unit MHz) can strengthen the robustness. The inset shows the $C_N$-not gate fidelity with ${\mathrm{\Omega }}_{\max }/2\pi =1$ MHz and $T_a=10\mu \mathrm{K}$, which remains high when $N=5$ [82, 84]. Each point is an average of 201 realizations.

Figure 8

Schematic diagram for the two-photon Rydberg excitations in ${}^{87}\mathrm{Rb}$ atoms. ${\sigma }^{\pm }$ denote the polarization of lasers.

Figure 9

Fidelity of performing a not gate on an initial state $(|0{⟩}_1-|1{⟩}_1)\otimes (|0{⟩}_t-|1{⟩}_t)/2$ based on the four-level atoms shown in Fig. 8 for examining the validity of the two-photon processes by using the OHQC scheme. The inset shows the fidelity towards the end of the gate.

Figure 10

(a) Error sensitivity $S$ versus $a_1$ and $a_2$. Blue curves are 0.5 contour lines of $S$. (b) $Tmax[{\mathrm{\Omega }}_t(t)]/2$ versus $a_1$ and $a_2$. Black dashed curves are contour lines of $Tmax[{\mathrm{\Omega }}_t(t)]/2$ with different values. (c) Pulse forms with $max[{\mathrm{\Omega }}_t(t)]/2\pi =1$ MHz, $a_1=0.28$, and $a_2=-0.12$.

Figure 11

Trace-preserving-operator-based average fidelities of controlled ${\hat{U}}_t(\theta ,\varphi ,\gamma )$ gates with different values of $\gamma$. (a) Two-qubit gates. (b) Three-qubit gates. $\tau =400\mu \mathrm{s}$.

Figure 12

(a) Level structures of a pair of coupling control and target superconducting qubits by resonant exchange interaction with strength $J_0$. (b) Level structures of two coupling control qubits by nonresonant exchange interaction with strength $J_0^{\prime }$ and detuning (anharmonicity) $\delta$. The decoupled state $|D⟩$ for each qubit is omitted for simplicity.

Figure 13

(a) Fidelity versus $J_0^{\prime }/{\overline{\mathrm{\Omega }}}_c$ and $\delta /{\overline{\mathrm{\Omega }}}_c$ for performing a Toffoli gate. $J_0=\omega =200{\overline{\mathrm{\Omega }}}_c$, ${\overline{\mathrm{\Omega }}}_c=30max[{\mathrm{\Omega }}_t(t)]$. (b) Fidelity evolution of performing the Toffoli gate. $J_0=J_0^{\prime }=\omega =2\pi \times 50$ MHz, $\delta =2\pi \times 400$ MHz, ${\overline{\mathrm{\Omega }}}_c=2\pi \times 7.5$ MHz, and $max[{\mathrm{\Omega }}_t(t)]=2\pi \times 0.25$ MHz.